DSpace at VNU: Multipliers for Generalized Entire Dirichlet Sequence Spaces

- 1999

VNU. JOURNA. OF SCIENCE. (Mat. Sci., t .x v

M U L T IP L IE R S
e n t ir e

FOR

D IR IC H L E T

G E N E R A L IZ E D

s e q u e n c e

sp a c e s

T rin h D ao C h ie n
Girl L hi Ed u c Ht i o i i i ìi ìd Tvri i i i i ng DepHi ' t I i i ent

I. I N T R O D U C T I O N

Given a ipquencc (ÀA-) with A^. G c , 0 < |Aa.| t +00 and p > 0 , consider tho gpneraliz'i
entire Dirichbt series
oc

^CA.£'p{Ai.z), 2 G C .
k=\
where coefficents

(1.:

an' fomplpx nunibers and Ep[z) is tho Mittag-Lefflei funct ion:
( r being the G a m m a function).

, f r o r ( i + i)

In [5] W( piovfd that if the spiios ( 1.1) convorges absolutely for all 2 G c
log In..
l in i s u p - ^ f —-- = - 00.
a n d conversev, if t H. coofficient.s of the s.n ios ( 1. 1) satisfv coiiditioii ( 1.2 ) and if

lo^A‘
|Aa-

iiinsup —

< -1-00

flien the serũs ( 1.1 couvvv^cs absolut(‘ly for all z e c
N e x t , i n t h e C
t h . ' r r , n . l it io , i (I :i)

i„

car,., o f [2], w<-

t h e followingsoquouí' space
4 = { q . : ca- satisfies (1.2)} = {(r^.): lini s u p | a . |
A--^oc

D e n o te d by

tli( Kothe d u a l of A. i.e..
CXj

= {("a ) : ^

< + 0 0 for all (f>) e A}.

|r/,.

Ả-=1
we proved thit A ‘^ = c.

= A . Honco C" = A . whc-io

c = |(i/A.) : lim Slip
A--00
Furthenioip,

each c = (c*.) e A. wo dpfined
A = sup Ct i /IAaT

*■>1

< + 00Ị

-I

I" = 0}


Ti'inh ha o C h i e ĩ i

9

Iu [5], 1)V us in g t h e saiiu' i n r t h o d as iu [2], \V(' Ị)rovO(l tliat

Ả is a C()iu])l('t(' s('pariHlt)l('.

noii-iiormablc. nio ti iz a h lr space, whoK' th(' iiiPtiic is j;iv('u t),v

( ỉ I > ) = II" - ^’11.4 ; " = ("/. ) e
I n t l i i s n o t e . VV(' c o i i t i i n H ' t o s t u d y m u l t i p l i i ' i s t x ' t w i ' c n t h e s o s p a c e s a n d ot l i M s e q u i c n c i '

spaces oil spaces A ami c.
w v K'call th at for two S('qu(nic(’ spaces A' and y . tlu' symbol (.V. Y ) -li'iiotrs the
s('(ỊU('ncp s p a c e o f m u lt i p li e r s from A' t o i
(A'. Y ) =

P.R.. [1]).

(a"A-) e 'i' foi all ( a . ) € A'} .

It is obvious tliat if

A'l c A\> and V'l c Yo- tlieii (X->- i ' l ) c (A'l, V'i).

(1.4)

Also it is clear t h a t , th e K o t h c du a l of a soqiiencp space is, in fact, th e soqtLK'iK'c
spaco of mu ltipliei s from this space to /, i.e.. (-4, h) = A ' \ A qupstion arises: wliat aibout

multipliois from A and c to I,, (0 < p < +oo) and vifc-veiHH'.' This is the sub ject i:)f the
p m se n f note.

1 would like to expif'ss my deep g r a t i t u d e to Prof. XRuyt-ii Van

M a n a nd I)i

L.f' Flai

Khoi for helpful stiggostioiis ill th e p r e p a r a t i o n of thi s papol,

II. MULTIPLIERS FOR GENERALIZED ENTIRE rJIRICHLET S E Q U E N C E sp .- \c?;s
First. \V(‘ not(' tli(' followiui> li'sult

L e m m a 2.1. U e hrive
A d i , c l o o C CẢ) < p < + 00.
Wo provo the following l(>ninia.s
L e m m a 2.2. W'e lìỉìvc
a)
C)cC.
l>)Cc

A),

c) c c (C„ C).
Proof:

a) Lf't (i/A.) e {A. C). Sr.pposo th at (»A.) ệ c. T h e n for a ib itra ry M > 0 a n d foi a
scquriicp (f,,), 0 < e„ i 0, tlK'io exists an iiicrea.sing scquencf' (A-„) of positive
such t h a t
>

M -£ n .

V/; > 1.

We defiue th e sequence (c^.) as follows

nk
Ck =

0,

1/2

if

othorwise.

n -

1, 2 ,...,


M u ltọ lie r s f o r Generalizec E n tire ...

3

lio n Wf* liaví'
1/2

<
;1I11sup(A/

0 , ay M

= A/

n —oc
-oc

+00

S' ( a - ) G A. How ev or. we lavo

liin sup |r*.u.|’/ 1-^*1'’ = limsup(|i/A.

>

Ì1—»oo^

A--»oo
liinsup(A / - f

-> oo.

n—oc

/

as

M

+OŨ .

This iiplies that (Cf,. m.) ệ c w iid i leads to a rontradictioii.
T i p implications b) and c) aiP obvious
□
N i w WP c a n p r o v e t h o f o l l c w i i i g r e s u l t

T h e o e n i 2 . 1. We have

(^ . 0 = (/,„ C) = (/^ , C) = (C, C) = ( ^ ,

= ( ^ , Ì,,) = ( A /oo) = c.

PTOof.Vvom Loiiiina 2.1, Lenur.a 2.2 ami (1.4), it follows th at

c {A. I,) c ( A loc) c {A, C) c c,

C’ c ( A
and

C’ c {C, c c
T f tli(*oiPin

is proved

(/oc.

C) c (/;„ C) c ( A C) c c .

c

we prove the followiiiJL e r n n i 2.3. We ỈIỈÌVC

a)'/,,. .4) c .4,
hj C, U ) c A .
c)Ac

i C,

A).

Proof:
a)-irst, we Iiotp th at (q.) E

a

ÌỈ and only if {(ị) G A (with any appropriato choicp

of the ow n ). Furthoi more, we can check th a t the soquencp (Aa-) satisfies Condition (1.3)
if and aily if tlioir exists a > 0 such that
CXD

< + 00.

(2 . 1)

*-=1
Nw. let (;/*.) e {Ip, A). Suppose th a t (i^.) Ệ A , which means th a t {uị) ị A . Then
there eists M > 0 such th a t for a sequence (£„) ị 0 , there exists an increasing sequence
(A:„) of)ositivo numbers such that


T ỉ-inh D a o C h i i e n

> M - £„ , V7Í

Akn
i.

> 1.

This implies th a t
{£„ - A/)|Aa„

, Víí > 1 .

Dofinp a sequoiicp (r^.) as follows:
exp

if k = k'n, ÌÌ — L 2,

,

Ck =

ot herwiso,

0,

where 7 < i\I — a and a > 0 is defiiKHl by (2.1). Then, we have
Ck

^ e x p

oc

oc

z_>
__1

i—J

'( 7 - i n ) aa-„

r]

<

_1

[(7 -

< +00,

Tl= i

77=1

due to (2.1). which shows th a t (ca-) € Ip. However,

lini sup

log

Aa.

k—*oc

= lira sup

log f'A-„ »A'„ p
= lim s u p (7 - e„) = 7 > - o o ,
n—*oo
Aa-„ p

which moans th at ịịct,-ÌIk-)’’) Ệ Ả or {ckiik-) ị Ả. This is a contradiction. Hf“iìC(’ {Ij,, ^ 4 ) c

b) Let (iu.) G (C, /oo)- Assume th a t (?/.*.) ị A , then t h n c exists au increasing s vq n fv m v
(Ả'„) of positive' Iiuiiihois suc’li tliat
lilll |»A.„ I

"

n —►
OC'

= + 00 .

( (

Consider a soquonce (aO as follows:
f A -„/|ỉ/a-J,

Cị,- — <

[ 0,

if />■

A-;,,

ÌÌ =

1, 2 , , . . . ,

othowiso.

Then we have
= 0 < +00 .
Ả—*oo

A-—*oo

duo to (2.2) and (1.3). Hence (ct) G c. Howrvor
sup | c A . i/A-|
A'> 1

=

sup l a - , , i/j.,, I = sup h„
ÍÍ > 1
T7> 1

Hence {Ckĩỉk) ị loc- a contradiction.
c) T he implication A c (C, A ) is obvious.
We can prove the following

□

=

4-00 .

2.2 )


M'dtipliers f o r G e nera lized En tire ...
T l e o r o m 2.2.

5

W’v liiiv(>

(C. 1^ ) - {C. I„) = (C.

(/^ .

^ ) = .4.

P tio/: Fioiii l .ci iim a 2.1. l.ciiiiiia 2,:i a n d ( 1, 1) . it follows t h a t

^ c {C. A) c {1^. A) c
Til' ĩll('Ulf'ni is
R f i n a i k.

A) c A .

□

riKHjH'in 2.1 a n d 2.2 for i h ( ‘ (Ji(linai \' Dil iclil('ĩ s('i i('s OÍ 0 1 H' a n d sf'\'(Mal ('(Jiiiplrx

varal)l('s wTvr ỊM‘0V(‘(1 in

a n d [4ị.

R EFER EN C ES
[1 J . M . Ai iil crs ou cV A . L . Shields.

C'cx'fficient I iiu lli pl ic is of B l o c h f u n c t i o n s ,

Tnnis.

Aiiier. Math. Soc. 2 2 4 ( 1 9 7 6 ) . 255-205.

2 Lc Ilai Khoi. ilol oiiHHphic Diliclilot s e rie s iiis(>v(‘ial v a i i a h l c . Math. Scnriil

77(1995)

85-11)7.
3 Lc Hai k h o i . M u l t i p l i e r s for I^iiichlf't s c rie s in tli(' coiiiiili'x |)laii('. S o u t h - E a s t A s i a n
Mat h. Bull. ( Ĩ(J aỊ)Ị)(‘a i ' ).
;4 \ j ' Hai l \h o i .

Coefficient m u l t i p l i ( 'i s f(ji SOUK' classc.s o f Diriclilct s c ri e s ill s('C(nal

Í'()U1Ị)1(‘X \ariaỉ)l('s. A c ỉ a Mdi ỉ i .

\'ỉ( t ì i af i i ỉ ( ‘ti ( í o apỊ)('ai ).

5 l n n l . D a o ( ’lu cn. S c q u c u c c S])acc otCof'ffic'inith of o('iicializ(-(l (-ntiic Diiiclilct S('IÌ('S.

\ 'N Ư Journal o f S c i n u T . X a t . Sci., I . X I V . . \ o K 1 9 9 8 ) . 8-15.

TAF CHI KHOA HỌC DHQGHN, KHTN, t .x v , n‘’ l - 1999

M IA X r i ” CVA KHOXC; C;iAX 1)A\'
i) iH ! ( 'iií j: r XCỈUYKX s r v íU).\c:
l Y i n h Đ à o C h ìố ii
S ờ (Ỉiỉía (lục viì Đ à o ĨỈÌO (Ỉiỉì ĩ^ỉỉi

V(/] l.ai klioiip, ^ian (là\- A \'à V, khoii^ gian ílã>' cua các Iilian tir từ .V \'ào y \ kv
là ( A .) ). (lìrtrc x;íc (lịnh ỉilur sau (A '.)') := {(///.):
khuip ” ia:i (lãv A . cái' liọ su c ù a cliuỗi Diriclilct s u v r ộ n g (lạiio
Eậ

ị

G

V(V‘A-) G A'}. Xót

Ỹ2

íroi io (ló

là hàin M i r t a o - Lofflor. Q u a in o t ả klioiift nịaii A ' ' đ ố i n g ẫ u K ỏ t h e c ù a A . t a t h ấ y

ră:ií { A . l / } ~

troiio (tó /i = {(///^.);

|//^.| < oc}. M ộ t c á u hòi đ ặ t ra: kố t q u ả sõ

n hư thế nao (lối vái các khono »,ian (lãy n i a các Iihán từ fừ A . A ‘' vào rác khong «ian
qupi t h u ọ ' khác, c h ẳ i i - han /,,(0 < p <
đ ế n c á c noi (luiiíi íló.

o o ) , / o o . . . . và Iigirợc lại? Bài báo Iiày S(' đồ cập